1. Loeb Measures

Abstract

• 1.1 Introduction

• 1.2 Nonstandard Analysis

  • 1.2.1 The hyperreals

  • 1.2.2 The nonstandard universe

  • 1.2.3 -saturation

  • 1.2.4 Nonstandard topoloty

• 1.3 Construction of Loeb Measures

  • 1.3.1 Example: Lebesgue measure

  • 1.3.2 Example: Haar mesure

  • 1.3.3 Example: Wiener measure

  • 1.3.4 Loeb measurable functions

• 1.4 Loeb Integration Theory

• 1.5 Elementary Applications

  • 1.5.1 Lebesgue integration

  • 1.5.2 Peano’s Existence Theorem

  • 1.5.3 Itô integration and stochastic differential equations